Ab Initio Structural Energetics of Bare $\beta-$Si$_3$N$_4$ Surfaces and the Interface with Sm$_2$O$_3$

Motivated by recent electron microscopy studies on the Si$_3 $N$_4$/rare-earth oxide interfaces, the atomic and electronic structures of bare $\beta-$Si$_3$N$_4$ surfaces are investigated from first principles. The equilibrium shape of a Si$_3$N$_4$ crystal is found to have a hexagonal cross section and a faceted domelike base in agreement with experimental observations. The large atomic relaxations on the prismatic planes are driven by the tendency of Si to saturate its dangling bonds, giving rise to resonant-bond configurations or planar sp$^2$-type bonding. We predict three bare surfaces with lower energies than the open-ring ($10\overline{1}0$) surface observed at the interface, which indicate that non- stoichiometry and the presence of the rare-earth oxide play crucial roles in determining the termination of Si$_3$N$_4$ matrix grains. We also present preliminary first principles results for the Si$_3$N$_4$/Sm$_2$O$_3$ interface and compare our findings to experimental observations.